B1. The generating functional Z [J ]

The generating functional Z[J ] is a key object in quantum field theory - as we shall seeit reduces in ordinary quantum mechanics to a limiting form of the 1-particle propagatorG(x, x′|J) when the particle is under thje influence of an external field J(t). However, beforebeginning, it is useful to look at another closely related object, viz., the generating functionalZ(J) in ordinary probability theory.

Recall that for a simple random variable φ, we can assign a probability distribution P (φ),so that the expectation value of any variable A(φ) that depends on φ is given by

〈A〉 =

∫dφP (φ)A(φ), (1)

where we have assumed the normalization condition∫dφP (φ) = 1. (2)

Now let us consider the generating functional Z(J) defined by

Z(J) =

∫dφP (φ)eφ. (3)

From this definition, it immediately follows that the ”n-th moment” of the probability dis-tribution P (φ) is

gn = 〈φn〉 =

∫dφP (φ)φn =

∂nZ(J)

∂Jn|J=0, (4)

and that we can expand Z(J) as

Z(J) =∞∑n=0

Jn

n!

∫dφP (φ)φn

=1

n!gnJ

n, (5)

so that Z(J) acts as a ”generator” for the infinite sequence of moments gn of the probabilitydistribution P (φ).

For future reference it is also useful to recall how one may also expand the logarithm ofZ(J); we write,

Z(J) = exp[W (J)]

W (J) = ln Z(J) = ln

∫dφP (φ)eφ. (6)

1

Now suppose we make a power series expansion of W (J), i.e.,

W (J) =∞∑n=0

1

n!CnJ

n, (7)

where the Cn, known as ”cumulants”, are just

Cn =∂nW (J)

∂Jn|J=0. (8)

The relationship between the cumulants Cn and moments gn is easily found. We noticethat

∂W (J)

∂J=

∂ ln Z(J)

∂J=

1

Z

∂Z(J)

∂J∂2W (J)

∂J2=

∂

∂J

(1

Z

∂Z(J)

∂J

)=

[1

Z

∂2Z(J)

∂J2− 1

Z2

∂Z(J)

∂J

], (9)

and so on. Continuing in this way, and using eq.(6), we can find the {Cj} in terms of {gj},by expanding ln Z[J ] = ln

{exp

[∑n

1n!gnJ

n]}

, and comparing coefficients of Jn. We thenfind that

C1 = g1,

C2 = g2 − g21,

C3 = g3 − 3g1g2 + 2g31,

C4 = g4 − 3g22 − 4g1g3 + 12g2g

21 − 6g4

1; (10)

and so on.Finally, note that in simple probability theory, we can make a ”Legendre” transformation

to change the dependent variables; if we write

W (J) = Γ(φ) + Jφ, (11)

then in terms of the new function Γ(φ), we have

∂Γ(φ)

∂φ= −J. (12)

and we can then rewrite everything in terms of Γ(φ) and its derivatives.We will see in what follows that all of these functions of simple probability theory can

be put into correspondence with functionals in QFT. As we will also see, it is useful andilluminating to explore the analogies between probability theory, field theory, and thermo-dynamics/ statistics mechanics.

2

B.1.1: GENERATING FUNCTIONAL and PROPAGATORSfor SCALAR FIELD THEORY

Suppose we are now dealing with a quantum field φ(x) in D dimensions. Then the ana-logue of the generating function in eq.(3) for ordinary probability distribution is a generatingfunctional Z[J ], given by

Z[J ] = N

∫Dφ A(φ; J = 0) ei

∫dDxJ(x)φ(x)

= N

∫Dφ e

i~S[φ] ei

∫dDxJ(x)φ(x), (13)

and we see that the analogue of the probability distribution P (φ) is the ”partition function”Z[J = 0, φ], i.e., the normalization factor N in eq. (13) is

N−1 = Z[J = 0] =

∫Dφei/~S[φ] ≡

∫DφA(φ),

A(φ) = ei/~S[φ] = ei/~∫dDxL(φ(x)), (14)

so that the analogue of the probability distribution P (φ) for a random variable to take thevalue φ is now the amplitude A(φ) = ei/~S[φ] for a field to take a given configuration φ(x).

We shall see later what we must write to deal with probabilities (as opposed to ampli-tudes) in QFT, using a kind of ”double path integral” technique (recall that in ordinary QMone discusses probabilities using the density matrix, which also can be understood in termsof a double path integral).

Note also that a convergence factor has been left out of eqs.(13) and (14). Just as inordinary QM, we need to specify which configurations are being integrated over, and withwhich boundary conditions. In ordinary QM, this is done by rotating the end points of thetime integration to ±i∞, or by adding a term to the exponent to force the Green function∫Dqei/~S[q] to pick out the ground state (see notes on path integration for QM, in section

A). The same is done here - we write

Z[J ] = N

∫Dφ A(φ; J = 0) ei/~

∫dDx[J(x)φ(x)+iεφ2(x)]

= N

∫Dφ ei/~

∫dDx[L(φ(x))+J(x)φ(x)+iεφ2(x)]. (15)

I do not go through the demonstration that this gives the desired result, since it is similarto (but more complicated than) the demonstration for ordinary Q.M.. In any case, the keypoint here is that

Z[J ] = 〈0|0〉J(x), (16)

i.e., Z[J ] is the vacuum expectation value for the theory, under the influence of an externalsource J(x).

From now on all reference to the convergence factor in eq.(15) is suppressed.

3

Continuing now in analogy with ordinary Q.M., and with the simple theory of probability,let us define the time-ordered correlators

Gn(x1, · · · , xn) = 〈0|T{φ(x1), · · · , φ(xn)}|0〉 =

∫Dφ ei/~S[φ] φ(x1) · · ·φ(xn)∫

Dφ ei/~S[φ], (17)

where the denominator is added to normalize the result (we cannot in general assume theanalogue of eq.(2)); the corresponding power series expansion is then (cf. eq.(5))

Z[J ] =∞∑n=0

(i/~)n1

n!

∫dx1 · · ·

∫dxnGn(x1, · · · , xn) J(x1)J(x2) · · · J(xn), (18)

and the Gn is related to Z[J ] by

Gn(x1, · · · , xn) = (−ih)nδnZ[J ]

δJ(x1) · · · δJ(xn)|J=0 (19)

(cf. eq.(4)).Noting that Z(J) in ordinary probability theory is just the expectation value of the

function eφJ , we see that in scalar field theory, we have

Z[J ] = 〈0|T{ei/~∫dxJ(x)φ(x)}|0〉 (20)

which when expanded out gives

Z[J ] = 〈0|0〉 +i

~

∫dDx〈0|φ(x)|0〉J(x)

+1

2!

∫dDx1

∫dDx2〈0|T{φ(x1)φ(x2)}|0〉J(x1)J(x2) + . . . (21)

in agreement with eqs.(15) and (16).It is useful to represent the formal results diagrammatically. We represent the vacuum

functional Z[J ] as the diagrammatic eqtn. (22) which is just the series sum in eqtns. (18)and (21); here we assume the Green functions are represented diagrammatically as thediagrammatic eqtn. (23)

These diagrams are drawn according to well-defined rules, which can be read off from therelevant equations. Thus we have the correspondences:

f(x) −→ × J ≡ i

~

∫dDxJ(x)f(x) (24)

4

where the cross indicates the external source current J(x), and this is connected by the line,or ”external leg”, to some function f(x). Thus we multiply whatever this ”external leg” isattached to, by the factor i

~J(x), and then integrate the combination over x. This is clearin eqs. (18) and (21). Note that whereas Gn(x1, · · · , xn) has n external legs, Z[J ] has none- they are all integrated over.

We get another useful relation, of recursive form, between Gn(y) and the generatingfunction Z[J ] by comparing the 2 sides of eq. (20) in series expanded form

G1(y) = −i~δZ[J ]

δJ(y)

= −i~ δ

δJ(y)

∞∑n=0

(− i~

)n1

n!

∫dx1 . . . dxnGn(x1, · · · , xn) J(x1) . . . J(xn) (22)

and reorganizing the labelling in the series, we get

G1(y) =∞∑m=0

(i

~

)m1

m!

∫dx1 . . . dxmGm+1(x1, · · · , xm; y) J(x1) . . . J(xm) (26)

which is represented as (27) (adding an external leg to each side of eq. (26)): It is an equationto which we will return.

Note that this corresponds to a very simple result in ordinary probability theory; fromeq. (5) we have

g1 = 〈φ〉 =

∫dφ P (φ)φeφJ =

∞∑n=0

Jn

n!φn+1P (φ) =

∞∑n=0

1

n!gn+1J

n (28)

So far everything we have said has been true for the Lagrangian density L[φ(x)], i.e., wehave not used a specific interaction like a φ4(x) term. This will come later - first we have todevelop more of the formal theory.

5

B.1.2: CONNECTED GREEN FUNCTIONS for SCALARFIELD THEORY

We have seen how the generating functional or partition function or vacuum expectationvalue Z[J ] corresponds to the generating function Z(J) for simple probability distributions.Let us now continue this analogy, and write

Z[J ] = ei~W [J ] (29)

(composes eq. (6)). We now proceed in complete analogy with eqs. (6)-(10), by definingwhat we call ”connected” Green functions (the name will become clear) as

G(c)n (x1, . . . , xn) = (−i~)n

1

Z[J ]

δnZ[J ]

δJ(x1) · · · δJ(xn)|J=0 (30)

and

W [J ] = −i~∞∑n=1

(i

~

)n1

n!

∫dx1 . . . dxn G(c)

n (x1, · · · , xn) J(x1) . . . J(xn) (31)

Composing eqs. (19) and (30), we see that

Gn(x1, . . . , xn) = Z[J ] G(c)n (x1, . . . , xn) (32)

We see that the connected Green functions here are the analogue of the ”cumulants” thatwe defined for the ordinary probability distributions - the 2 key differences being

(a) we now use W [J ] = −i~ lnZ[J ] instead of W (J) = lnZ[J ], since we deal with thequantum mechanics(see footnote 1 below); and

(b) we now deal with functionals instead of functions.

Actually the closer analogy between Z[J ] and probability theory is with the ”character-istic function” Z(J) (as opposed to Z(J) in eqs. (1)-(9))

Z(J) =

∫dφP (φ)eiJφ = 〈eiJφ〉 (33)

1NOTE: It is common in the literature to write Z[J ] = eW [J], in very close analogy with eq. (6). Wethen have the definitions and results as follows:

W [J ] =

∞∑n=0

(i

~

)n1

n!G(c)n (x1, . . . , xn) J(x1) . . . J(xn)

G(c)n (x1, . . . , xn) =

1

Z[J ]

δnZ[J ]

δJ(x1) · · · δJ(xn)

1

Z[J ]

δZ[J ]

δJ=

δW [J ]

δJ

G1(y) = −i~δZ[J ]

δJ(y)= Z[J ] G(c)

1 (y) = −i~Z[J ] G(c)1 (y)

6

which can also be thought of as a simple Fourier transform of P (φ). To differentiate betweenall of these, we have put a bar over quantities like W or Z to indicate functions that are theresults of simple exponentiation (like Z(J) = eW (J)) as opposed to Fourier transformation

(like Z(J) = ei~W (J)).

Now let’s consider the relation between the connected Green functions and their graphicrepresentation. Notice first of all from eq. (31) that we can make some sort of graphicexpansion for W [J ] as we did for Z[J ], i.e., (34) where the connected Green functions areshown as solid circles; and as before we can write (cf. eq. (26)):

G(c)1 (y) = −i~δW [J ]

δJ(y)=

∞∑m=1

(i

~)m

1

m!

∫dx1 . . . dxm G(c)

m (x1, · · · , xm; y)J(x1) . . . J(xm) (35)

or, in diagrams, we have (36)So far so good. But now let’s look at how we represent eq. (32) graphically; we just have

the relation (37). How should we interpret the result in (37)? The left-hand side is the setof all graphs with n external legs (connected to the points x1, . . . , xn). On the right-handside we can then imagine taking all the n-leg graphs, and separating out all those that areconnected graphs, i.e., all those that cannot be separated into 2 parts. Then the set of alln-leg graphs must be made from this connected set, multiplied by those parts disconnectedfrom them. But this set of all disconnected parts has no external legs, and is therefore justZ[J ]. Thus we get (37).

7

If this argument seems too glib to you, a more rigorous algebraic argument is given asthe end of this section, in which the exponential in eq. (29) is translated directly into asum of products over connected graphs. An even more transparent discussion appears in thesection on perturbative expansions and ”diagrammar”.

B.1.3 VERTEX FUNCTIONS for SCALAR FIELD THEORY

So far we have expanded two different functionals as infinite series, i.e., the vacuumenergy generating functional Z[J ] (the partition function), and the generating functionalW [J ] for connected diagrams (analogous to the free energy in statistics mechanics), relatedby

W [J ] = −i~ lnZ[J ] (38)

with expansions

Z[J ] =∑n

(i

~

)n1

n!

n∏j=1

∫dDxj Gn(x1, . . . , xn) J(xj)

W [J ] =∑n

(i

~

)n1

n!

n∏j=1

∫dDxj G(c)

n (x1, . . . , xn) J(xj) (39)

Now we introduce a 3rd expansion, in terms of a new generating functional Γ[φ]; this however,is not a functional of the external currents, but of the field φ(x) itself. We write

Γ[φ] =∞∑n=1

1

n!

∫d4x1 · · ·

∫d4xn Γn(x1, . . . , xn) φ(x1) . . . φ(xn) (23)

so that

Γn(x1, . . . , xn) =δnΓ[J ]

δφ(x1) · · · δφ(xn)|φ=0 (40)

Now of course the way to change variables between the J(xi) and the φ(xi) is by aLegendre transformation, with form given here by

Γ[φ] = W [J ]−∫d4xJ(x)φ(x) (41)

Thus now we now have the pair of relations

δW [J ]

δJ(x)|φ = φ(x)

δΓ[J ]

δφ(x)|J = −J(x) (42)

8

Later on we will look in some detail at vertex functions like Γn(x1, . . . , xn), which turn out tobe central to the approach of QFT to the dynamics of real physical systems - they are usedin everything from scattering theory to particle physics to transport theory in condensedmatter systems.

Here we will simply establish the relationship between the connected Green functionsG(c)n (x1, . . . , xn) and the proper vertices Γn(x1, . . . , xn). Let’s first do this for the simplest

case, that of the 2nd-order correlators G(c)2 (x1, x2) and Γ2(x1, x2). From their definitions, and

from eq. (42), we have

G(c)2 (x1, x2) = −~2 δ2W [J ]

δJ(x1)δJ(x2)|J=0 = −~2 δφ(x1)

δJ(x2)|J=0

Γ2(x1, x2) =δ2Γ[J ]

δφ(x1)δφ(x2)|φ=0 = −δJ(x1)

δφ(x2)|φ=0 (43)

from which we immediately find that∫d4y G(c)

2 (x1, y) Γ2(y, x2) =δφ(x1)

δφ(x2)= ~2δ(x1 − x2) (44)

whose Fourier transform is

G(c)2 (p,−p) =

−~2

Γ2(p,−p)(45)

i.e., the two functions are inverses of each other.To understand this result a little better it is helpful to look at it diagrammatically. Let’s

consider what kind of graphs make up G(c)2 (x1, x2). As we see above, the G(c)

n (x1, . . . , xn)are ”connected graphs”, which cannot be separated without cutting internal lines. Here weshow some in eqtn. (46).

We now notice that we can divide graphs like those above into 2 kinds. There arethose graphs which, apart from their ingoing and outgoing lines, cannot be separated into 2parts by cutting any internal lines - or to put it differently, they can be viewed as ”internalirreducible ” parts, connected to 2 external legs. Note that all of these ”irreducible vertexparts” are different from each other. Then we have graphs that can be separated into 2 partsby cutting an internal line; these are called ”reducible” graphs. We see that we can get allgraphs for G(c)

2 (x1, x2) by simply stringing together the irreducible parts in all possible ways.Let us put this formally; we define the following 2 quantities shown in (47), where G0(k)

is of course just the free propagator, which we have already met in our discussion of 1-particle

9

Green functions in QM. The diagram for −i∑

(k) has no external legs - the dashed linessimply indicate where they would attach to −i

∑(k). We work here in 4-momentum space,

simply because momentum is conserved. Now we obtain the full G(c)2 (k) ≡ G(c)

2 (k,−k) as(48) and so we have

G(c)2 (k) =

i~k2 −m2 − ~

∑(k)

Γ2(k) = −i~[k2 −m2 − ~

∑(k)]

(49)

and we see that the vertex part, apart from the term (k2 − m2) can be identified with−i∑

(k), which is of course just the self-energy.

One can continue this analysis for the higher-order functions G(c)n (x1, . . . , xn) and Γn(x1, . . . , xn).

Without going through the derivation, we find that

G(c)3 (x1, x2, x3) = −

∫d4y1

∫d4y2

∫d4y3 Γ3(y1, y2, y3) G(c)

2 (x1, y1)G(c)2 (x2, y2)G(c)

2 (x3, y3)

Γ(c)3 (x1, x2, x3) = −

∫d4y1

∫d4y2

∫d4y3 G(c)

3 (y1, y2, y3) Γ(c)2 (x1, y1)Γ

(c)2 (x2, y2)Γ

(c)2 (x3, y3)

(50)

as shown in the diagrams of (50). One can continue to Γ4, which can be written in terms of

G(c)4 ,G(c)

3 ,G(c)2 , Γ3 and Γ2; and so on; all the Γn(x1, . . . , xn) have the same ”tree” structure.

We have now written expressions for Z[J ] in terms of 3 different sets of quantities, i.e., the

correlators Gn(x1, . . . , xn), the connected correlators G(c)n (x1, . . . , xn), and the proper vertices

Γn((x1, . . . , xn). Now it turns out that each of these has an important physical meaning, butto better appreciate that we have to first go on with our formal developments.

10

B.1.4: FREE PROPAGATORS for φ-FIELD

Before we ever look at the effect of interactions on the field dynamics (and this is ofcourse the whole point of QFT), we need to establish the properties of the free field, withoutinteractions. What this means, as far as we are concerned, is determining the form of thecorrelation functions. Thus we will begin again with the generating functional Z0[J ] for theφ-field scalar theory, given in eqs. (13) and (15) above. Let us write this out explicitly forthe free system, i.e.,

Z0[J ] =

∫Dφ e

i~∫dDx[L0(φ)+φ(x)J(x)+iδφ2(x)]∫

Dφ ei~∫dDx[L0(φ)+iδφ2(x)]

(51)

where we include the convergence factor iδφ2(x) in the exponent, so as not to forget it. Nowactually we can evaluate eq. (51) exactly because the functional integrals involve quadraticforms. To make this more explicit, let’s rewrite the free field action, noting that (NB:∂2 = ∂µ∂µ is often written as � or ∆):∫

dDx (∂µφ∂µφ−m2φ2) =

∫dDx

[∂µ(φ∂µφ)− φ∂2φ−m2φ2

]=

∮dD−1x (φ∂µφ)−

∫dDx φ(∂2 +m2)φ (52)

using the D-dimensional variant of Gaussian theorem; here∮dD−1x denotes a surface inte-

gral. Thus we have the alternative form for Z0[J ] as

Z0[J ] =

∫Dφe−Q

0J (φ)∫

Dφe−Q0(φ)(53)

where we drop the total derivative ”surface integral” term in eq. (52), simply assuming that

11

|φ| → 0 fast enough at the boundaries of whatever system we deal with; and where we define

Q0(φ) =1

2

∫dDx φ(x)

[i

~(∂2 +m2)

]φ(x) ≡ (φ,Q0φ)

Q0J(φ) =

∫dDx

{1

2φ(x)

[i

~(∂2 +m2)

]φ(x)− i

~J(x)φ(x)

}≡ (φ,Q0

Jφ) (54)

These are standard Gaussian functional integrals, and we immediately get∫Dφ e−Q0(φ) = | detQ0|−

12∫

Dφ e−Q0J (φ) = | detQ0|−

12 e

12

(η,Q−10 η) (55)

where Q0 = i~(∂2 + m2), and η = i

~J(x). The determinants cancel in eq. (53), and we areleft with

Z0[J ] = e−i2~

∫dDx

∫dDx′J(x)∆F (x−x′)J(x′) (56)

where the ”Feynman propagator” is

∆F (x) = −(∂2 +m2 − iδ)−1 (57)

i.e., it is defined by(∂2 +m2 − iδ) ∆F (x) = −δ4(x) (58)

where δD(x) is the D-dimensional δ-function. Thus we have reduced the generating func-tional Z0[J ] to a simple function involving integrals over J(x) and ∆F (x− x′).

The Feynman propagator ∆F (x − x′) is the object that ties together the currents J(x)and J(x′), and it is therefore important to understand its properties. Since the free fieldsystem is translationally invariant, we Fourier transform it to get

∆F (x) =

∫d4k

2π∆F (k)e−ikx =

∑k

e−ikx∆F (k) (59)

where

∆F (k) =1

k2 −m2 + iδ=

1

k20 − (|k|2 +m2) + iδ

(60)

where the last form we use the Lorentz invariance of the theory. From this last form we cansee the pole structure of the theory in the frequency domain (i.e., in k0-space). Let’s seewhat this means when we come to integrate over k-space, as defined by eq. (59). If we dothe frequency integral first, then we must follow the path shown in the left of fig. 11 - to seethis, just rewrite eq. (60) as

∆F (k) =1

2k0

[1

k0 − (εk − iδ)+

1

k0 + (εk + iδ)

](61)

12

whereε2k = (|k|2 +m2) (62)

The contour can also be taken as shown in the right of fig. 11, if we desire to put the polesat ±(εk − iδ) onto the real axis (letting δ = 0). Note that in both cases we are simply usingartificial devices to deal with the branch cut structure of the propagator - something we havealready seen in ordinary QM (in section A). Let’s now look at what we get for the correlationfunctions in this free field theory. The simplest thing to do is to simply expand the functionin eq. (56), to get

Z0[J ] = {1 +∞∑n=1

1

n!

(−i2~

)n [∫dDx

∫dDx′J(x)∆F (x− x′)J(x′)

]n} (63)

a series which is obviously better written in terms of the Fourier transformation of J(x) and∆F (x); it is easy to show that we get

Z0[J ] = {1 +∞∑n=1

1

n!

(−i2~

)n [∑k

Jk∆F (k)J(x′)Jk

]n} (64)

and we see that both of these expressions have obvious diagrammatic versions; we show thisin (65) where the notation is pretty much as used in eq. (24); we have the diagram in (66)with integration over repeated indices (in this case the momenta).

To obtain the correlation function we go back to the exponential form in eq. (56), andjust functionally differentiate. Let’s do the function G2(x, x′) in detail. From eq. (19) we

13

have

G(0)2 (x, x′) = −~2 δ2Z0[J ]

δJ(x)δJ(x′)|J=0 (67)

Now, using the simple result

δ

δJ(x)e−i2~

∫dx1

∫dx2J(x1)∆F (x1−x2)J(x2) =

−i~

∫dx1∆F (x1 − x2)J(x1) e

−i2~

∫∫J∆F J (68)

We easily find thatG

(0)2 (x, x′) = i~∆F (x1 − x2) (69)

By continuing this analysis we easily find that G(0)3 (x1, x2, x3) = 0, and that

G(0)4 (x1, x2, x3, x4) = ~4∆F (x1 − x3)∆F (x2 − x4)

+ ~4∆F (x1 − x4)∆F (x2 − x3)− ~4∆F (x1 − x2)∆F (x3 − x4) (24)

whose diagrammatic interpretation appears as (71). More generally we have

G(0)2n+1(x1, . . . , x2n+1) = 0 (72)

for correlators with odd numbers of legs, and

G(0)2n (x1, . . . , x2n) =

∑p

G(0)2 (xp1 − xp2) · · ·G

(0)2 (xp2n−1 − xp2n)

= in∑p

∆F (xp1 − xp2) · · ·∆F (xp2n−1 − xp2n) (73)

for correlators with even numbers of legs; here∑

p means the sum over all the permutationsof the indices p1, p2, . . . , p2n. This last result is actually an example of ”Wick’s theorem”,which in the older canonical formulation of field theory, using field operators and theircommutation relations, is actually quite clumsy to prove.

So far what we have done is fairly formal. It is now time to turn to a specific theory, andtake a quick glance at two of the recipes used to deal with interactions in this theory, viz.,the φ4 theory.

14

B.1.5 INTERACTIONS and Z[J ] in φ4 THEORY

We have seen that the free field theory is simple to solve because we only have to deal withGaussian integrals. This feature disappears as soon as we introduce interactions. However,we may use the some techniques to deal with these as we did for the simple 1-particle Greenfunction for non-relativistic particle mechanics. What we will do in the following is to derive2 results for the full generating functional Z[J ] in φ4 theory. The derivations will be shortbecause they parallel very closely those in ordinary QM.

RESULT 1: FUNCTIONAL DERIVATIVE δδJ(x)

Let us recall the simple result for functional integrals, discussed in the Appendix andalready used in eq. (69) of the section on correlations in QM, that for a quadratic functionalQJ [φ] for the field φ(x) and a polynomial P [φ(x)] in φ(x), we have∫

Dφ(x)P [φ] e−Q[φ] = P (− δ

δJ)

∫Dφ(x) e−Q

0J [φ] (74)

where Q0J [φ] has the general form

Q0J [φ] =

i

~(φ,Q0φ) + Jφ+ C (75)

We now assume that the polynomial in question is just the exponential functional over theinteraction term in the Lagrangian. Thus, suppose we have an action

S[φ; J ] =

∫dDx

{−1

2φ(∂2 +m2)φ− V (φ) + J(x)φ(x)

}(76)

where in the φ4 theory,

V (φ) =g

4!φ4(x) (77)

Then our polynomial is just the exponential, i.e.,

P [φ] = exp

{−i~

∫dDx V (φ(x))

}→ exp

{−i~g

4!

∫dDx φ4(x)

}(78)

so that, from eqs. (13), (14), and (74), we have the normalized Z[J ] in the form

Z[J ] =e−i~

∫dDxV (−i~ δ

δJ(x)) Z0[J ][e−i~

∫dDxV (−i~ δ

δJ(x)) Z0[J ]]|J=0

(79)

with Z0[J ] given by eq. (56); thus, for the φ4 theory, we have

Z[J ] =e−i~

∫dDx g

4!(−i~δ

δJ(x))4

e−i2~

∫dDx1

∫dDx2 J(x1)∆F (x1−x2)J(x2)[

e−i~

∫dDx g

4!(−i~δ

δJ(x))4

e−i2~

∫dDx1

∫dDx2 J(x1)∆F (x1−x2)J(x2)

] ∣∣J=0

(80)

15

which is a result that can be systematically expanded in powers of g, to give a perturbativetheory for the effects of the φ4 interaction. The same kind of expansion can then be givenfor the correlation functions, starting from the expansion for Z[J ]. We will look at this inour section on perturbation and diagrammatic expansions, applied to the φ4 theory (sectionB3).

RESULT 2: FUNCTIONAL DERIVATIVE δδφ(x)

As we already saw in dealing with ordinary QM, we can also transform expressions likeeqs. (79) and (80) into expressions involving functional derivatives with respect to φ(x), asopposed to J(x). This relies on the identity, discussed in the Appendix on functionals, givenby

f

[−i δ

δJ(x)

]g [J(x)] = g

[−i δ

δφ(x)

]f(φ)ei

∫dx φ(x)J(x)

∣∣φ=0

(81)

for 2 functions φ(x) and J(x). This means that we can write

e−i~

∫dDxV (−i~ δ

δJ(x)) Z0[J ] = Z0

[−i~ δ

δφ(x)

]e−i~

∫dDx[V (φ)−φ(x)J(x)]

∣∣φ=0

(82)

from which

Z[J ] =Z0

[−i~ δ

δφ(x)

]e−i~

∫dDx[V (φ)−φ(x)J(x)]

∣∣φ=0

Z0

[−i~ δ

δφ(x)

]e−i~

∫dDxV (φ(x))

∣∣φ=0

(83)

i.e., we have converted the whole expression to one in which functional derivatives act directlyon the fields φ(x) themselves, rather than on the currents. For the φ4 theory with theinteraction in eq. (77), eq.(83) reduces to

Z[J ] =e−i2~

∫dDx1

∫dDx2 ∆F (x1−x2) δ

δφ(x1)δ

δφ(x2) e−i~

∫dDx[ g4!φ4(x)−J(x)φ]

e−i2~

∫dDx1

∫dDx2 ∆F (x1−x2) δ

δφ(x1)δ

δφ(x2) e−i~

∫dDx g

4!φ4(x)

∣∣φ=0

(84)

with again a perturbative expansion in g in the offing; both for Z[J ] and for the correlationfunctions.

B.1.6 PHYSICAL MEANING of THESE FUNCTIONS

We will only learn the full meaning of the various functions introduced here as we goalong, gaining experience with them and seeing how they are deployed. However we canlearn a lot by comparing what we have with other similar mathematical structures, and lookat their physical meaning. We can in particular look at the analogies with thermodynamicsand statistical mechanics, on the other hand, and with the ordinary problem of a single QMparticle in a ”noise field”, on the other.

16

In what follows we will (i) look at the analogies with thermodynamics and statisticalmechanics (and at the same time give a more complete discussion of connected graphs andW [J ]), and then (ii) compare results here with those for 1-particle Green function. For adiscussion of the relationship with noise in probability theory, go to the appendix on thistopic.

(a) CORRESPONDENCE with STATISTICAL MECHANICS

At the very beginning of the section, we saw the simple way in which the structure ofordinary probability functions corresponds to the generating functional of QFT. We canmake this correspondence more precise in one of two ways, i.e.,

(i) Reduce the QFT to a simple theory of functions by taking the classical limit, i.e., byrestricting ourselves purely to the classical solution φ(x) that minimizes the action.

(ii) By generalizing the simple probability theory given earlier, from a theory dealing withprobabilities of outcomes P (φ) for some random variable φ, to probabilities for outcomesP [φ(x)] of some random process φ(x), itself a function of some variable x. The outcomesP [φ(x)] are functional of φ(x), whereas P (φ) is a function of φ.

The correspondence with statistical mechanics or thermodynamics comes because thesesubjects deal precisely with probabilities for outcomes. Ordinary thermodynamics dealswith simple functions like the free energy F (T ), a function of temperature T ; statisticalmechanics, on the other hand, deals with functionals like the free energy functional F [φ]of some configuration φ(x) = φ(r, t) of the system. In a non-relativistic condensed mattersystem this configuration could be written in terms of the quantum field describing thesystem, with probabilities P [φ(x)] for different such configurations; or it might be writtenin terms of the probabilities P [ψj(x)] for the system to be in different eigenstates of theHamiltonian - it amounts to the same thing. In a relativistic QFT we will always deal withfield configurations.

In what follows we will make things simple by focussing on the classical limit. Later,when we have gotten used to the tools of QFT, we will look in a little more detail at thecorrespondence between QFT and full-blooded statistical mechanics.

The classical limit of a QFT for a simple scalar φ-field is defined by minimizing the action,i.e., for an action

S[φ, J ] = S[φ]−∫dDxJ(x)φ(x) (85)

We define the classical solution φJ(x) by

δS[φ, J ]

δφ(x)

∣∣φ=φ

=δS[φ]

δφ(x)

∣∣φ=φ

+ J(x) (86)

The generating functional Z[J ] then becomes a generating function, i.e.,

Z[J ]→ Zcl(J) = ei~ [Scl(φ)+

∫dDxφ(x)J(x)] (87)

17

where we have lost the functional integration over paths φ(x) because there is now only onepath φ(x), and the action S[φ]→ Scl(φ), for that path.

Now there is a very precise correspondence between the functions Zcl(J), Wcl(J), andΓcl(φ), on the one hand, and the thermodynamics function F (T ) or F (H) (free energy as afunction of temperature T or applied field H), the partition functions Z(T ) and Z(H), andthe functions U(S) (internal energy as a function of entropy S) and G(M) (thermodynamicsGibbs free energy as a function of magnetization), on the other hand. These links canbe summarized in the table (88): Thus we see that in these cases, the generator Wcl(J) ofconnected graphs corresponds to the free energy F (T ) or F (H), both functions of ”external”(and intensive) degrees of freedom, like J . On the other hand, a Legendre transformationto extensive degrees of freedom like S and M corresponds to the transformation of the fieldtheory to Γcl(φ), a function of the field itself, which is also an extensive variable.

This analogy between statistical mechanics and QFT is one we will pursue quite often,with increasing sophistication. But notice the key difference here, which is seen in thecorrespondence

i~→ kT (89)

between the two. This difference, between real and imaginary quantities, is in one way quitecrucial - probabilities in the statistics mechanical system correspond to amplitudes in theQM or QFT system. One can of course change this by rotating to imaginary time, to get aEuclidean QFT - we will also look at this later on.

We have already explored, in our discussion of 1-particle QM and correlation functions,the connection between G(0, 0|J), or Z[J ], for a particle in the presence of a ”noise source”J(t), and the path integral expression for this propagator. Without going into too manydetails (which we will do later), it is very revealing to look at the result for a harmonicoscillator subjected to a noise force J(t). As we saw, this is given by the AMPLITUDE(”vacuum correlator”):

Z[J(t)] = e−i2~

∫dt

∫dt′J(t)D0(t−t′)J(t′) (90)

where D0(t) is the simple SHO correlator. Now consider the expression which gives thePROPAGATOR for some random process ξ(t) to follow some particular ”path”. This is

18

described by a generating functional

Z[k(t)] =

∫Dξ(t)P [ξ(t)] ei

∫dt ξ(t)k(t) (91)

where the probability for a given ξ(t) is given, for a Gaussian random walk, by

P [ξ] = e−i2

∫dt

∫dt′ ξ(t) k−1

0 (t−t′) ξ(t′) (92)

where k−10 (t − t′) is some correlator. We see already how we can start to make connection

between random walks, constrained in one way or another, and QFT - again, we will explorethis more later.

(b) CONNECTED GRAPHS

Finally, let’s briefly sketch the connection between the functional W [J ] and the connectedgraphs - this also helps us understand the meaning of the functions and functionals we havedefined. Consider first the relationship between the set of all connected n-point graphs,which we have called G(c)

n (x1, . . . , xn), and the set of all graphs with n external legs, whichwe have called Gn(x1, . . . , xn). This relationship is intuitively rather obvious, and we havealready seen it shown graphically in (37). Clearly we can get all graphs for Gn(x1, . . . , xn)

by taking the set of all graphs for G(c)m1(x1, . . . , xm1), G

(c)m2(x1, . . . , xm2), . . . , G

(c)mp(x1, . . . , xmp),

such that∑p

j=1mj = n, and combining them in all possible ways, with all possible choicesfor m1, . . . ,mp summed over.

For those who do not trust the graphical argument given in (37), a direct calculationmay be more convincing. So let’s do this. Schematically we have the graphs in (93) where

we imagine graphs for G(c)m1 repeated qj times, and so on, and fix the condition that

p∑j=1

qjmj = n (94)

Now we just have a combinational problem. At order n, the number of different graphsis

Nn =n!

(m1!)q1q1! · · · (mp!)qpqp!(95)

19

and then summing over all n, we get for Z[J ] the expression

Z[J ] =∑n

(i

~

)n1

n!

∫dDx1 · · ·

∫dDxn

× J(x1) · · · J(xn)∑

q1m1+...qpmp=n

G(c)m1

(x1, . . . , xm1) · · · G(c)mp(x1, . . . , xmp) (25)

which we write as

Z[J ] =∑n

(i

~

)n ∑q1m1+...qpmp=n

p∏j=1

1

qj!

[∫dDx1 · · ·

∫dDxmj G

(c)mj(x1, . . . , xmj) J(x1) · · · J(xmj)

mJ !

]qj

=∑qj

∏j

(i

~

)qj 1

qj!

[∫dDx1 · · ·

∫dDxmj G

(c)mj(x1, . . . , xmj) J(x1) · · · J(xmj)

mJ !

]qj

= exp

[∞∑n=1

(i

~

)n1

n!

∫dDx1 · · ·

∫dDxn G(c)

n (x1, . . . , xn) J(x1) · · · J(xn)

](97)

so that, comparing with (31), we get again that

Z[J ] = ei~W [J ] (98)

which demonstrates again the relationship we were after.

20